Adiabatic response for Lindblad dynamics

We study the adiabatic response of open systems governed by Lindblad evolutions. In such systems, there is an ambiguity in the assignment of observables to fluxes (rates) such as velocities and currents. For the appropriate notion of flux, the formulas for the transport coefficients are simple and explicit and are governed by the parallel transport on the manifold of instantaneous stationary states. Among our results we show that the response coefficients of open systems, whose stationary states are projections, is given by the adiabatic curvature.Comment: 33 pages, 4 figures, accepted versio

Roughening transition, surface tension and equilibrium droplet shapes in a two-dimensional Ising system

The exact surface tension for all angles and temperatures is given for the two-dimensional square Ising system with anisotropic nearest-neighbour interactions. Using this in the Wulff construction, droplet shapes are computed and illustrated. Letting temperature approach zero allows explicit study of the roughening transition in this model. Results are compared with those of the solid-on-solid approximation

Boundary maps for C∗C^*-crossed products with R with an application to the quantum Hall effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t.Connes' pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schroedinger operators.Comment: to appear in Commun. Math. Phy

Time-Energy coherent states and adiabatic scattering

Coherent states in the time-energy plane provide a natural basis to study adiabatic scattering. We relate the (diagonal) matrix elements of the scattering matrix in this basis with the frozen on-shell scattering data. We describe an exactly solvable model, and show that the error in the frozen data cannot be estimated by the Wigner time delay alone. We introduce the notion of energy shift, a conjugate of Wigner time delay, and show that for incoming state $\rho(H_0)$ the energy shift determines the outgoing state.Comment: 11 pages, 1 figur

On the maximal ionization of atoms in strong magnetic fields

We give upper bounds for the number of spin 1/2 particles that can be bound to a nucleus of charge Z in the presence of a magnetic field B, including the spin-field coupling. We use Lieb's strategy, which is known to yield N_c<2Z+1 for magnetic fields that go to zero at infinity, ignoring the spin-field interaction. For particles with fermionic statistics in a homogeneous magnetic field our upper bound has an additional term of order $Z\times\min{(B/Z^3)^{2/5},1+|\ln(B/Z^3)|^2}$.Comment: LaTeX2e, 8 page

Complex Line Bundles over Simplicial Complexes and their Applications

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of Andr\'e Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension

Toward a unified theory of sparse dimensionality reduction in Euclidean space

Let $\Phi\in\mathbb{R}^{m\times n}$ be a sparse Johnson-Lindenstrauss transform [KN14] with $s$ non-zeroes per column. For a subset $T$ of the unit sphere, $\varepsilon\in(0,1/2)$ given, we study settings for $m,s$ required to ensure $$\mathop{\mathbb{E}}_\Phi \sup_{x\in T} \left|\|\Phi x\|_2^2 - 1 \right| < \varepsilon ,$$ i.e. so that $\Phi$ preserves the norm of every $x\in T$ simultaneously and multiplicatively up to $1+\varepsilon$. We introduce a new complexity parameter, which depends on the geometry of $T$, and show that it suffices to choose $s$ and $m$ such that this parameter is small. Our result is a sparse analog of Gordon's theorem, which was concerned with a dense $\Phi$ having i.i.d. Gaussian entries. We qualitatively unify several results related to the Johnson-Lindenstrauss lemma, subspace embeddings, and Fourier-based restricted isometries. Our work also implies new results in using the sparse Johnson-Lindenstrauss transform in numerical linear algebra, classical and model-based compressed sensing, manifold learning, and constrained least squares problems such as the Lasso

Piezoelectricity: Quantized Charge Transport Driven by Adiabatic Deformations

We study the (zero temperature) quantum piezoelectric response of Harper-like models with broken inversion symmetry. The charge transport in these models is related to topological invariants (Chern numbers). We show that there are arbitrarily small periodic modulations of the atomic positions that lead to nonzero charge transport for the electrons.Comment: Latex, letter. Replaced version with minor change in style. 1 fi